Exact expression for maximum Lyapunov exponent during transients in computationally powerful dynamical networks

TL;DR

This paper derives an exact analytical expression for the maximum Lyapunov exponent in a dynamical network, revealing how transient dynamics can be harnessed for computation and controlled through network parameters.

Contribution

It provides a novel exact formula for the maximum Lyapunov exponent during transients in complex networks, linking transient behavior to network structure and initial conditions.

Findings

Network exhibits positive MLEs during transients, enabling computation.

Analytical and numerical validation of the MLE expression.

Framework allows algebraic control of transient lifetimes.

Abstract

We study a network whose rich spatiotemporal dynamics have recently been shown to enable dynamics-based computation, including logic gates, short-term memory, and simple encryption. The network's time dynamics can be exactly solved through a nonlinear coordinate transformation. Here, we derive an exact analytical expression for the network's time-dependent maximum Lyapunov exponent (MLE). We demonstrate, both numerically and analytically, that the network exhibits positive MLEs during the transients that are useful for computation. Our framework enables algebraic manipulation of transient lifetimes through network connectivity and initial conditions, providing a rigorous theoretical foundation for understanding and controlling computation with transients.